The product of digits in the number 234 is 24: 2*3*4 = 24. Can you
describe a general procedure for figuring out how many x-digit numbers
have a product equal to p, where x and p are counting numbers? How
many different three-digit numbers have a product equal to 12? to
18? Why?

Before introducing children to the sieve of Eratosthenes, a parent seeks to reconcile
different methods found online. Doctor Peterson sees a didactic opportunity in the
discrepant steps: how many factors do you actually need to check?

Where is the first place that the difference between two consecutive
prime numbers exceeds 2000? Is there a formula or general approach to
finding such differences without having to just read through lists of
known primes?

I learned that a prime number was one divisible by only itself and 1,
but my 4th grader says that per her book a prime requires 2 different
factors. I note your Greek reference for 1 not being prime, which
would indicate that I'm wrong and there was no change in definition.
However, Ray's New Higher Arithmetic (1880) states, "A prime number is
one that can be exactly divided by no other whole number but itself
and 1, as 1, 2, 3, 5, 7, 11, etc." Can you tell me when this change
happened and why?

Just recently a grade six student asked me, "Why is 1 not considered
prime?" I tried to answer, but as usual, could not since I do not
understand this either. I thought the answer might lie in the fact that
we aren't using the true definition or we are interpreting it wrong.